On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius R₁ = 2.5 cm and winds its way out to radius R₂ = 5.8 cm. To read the digital information, a CD player rotates the CD so that the player’s readout laser scans along the spiral’s sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accurately adjust the rotational frequency ƒ of the CD as the laser moves outward. Determine the values for ƒ (in units of rpm) when the laser is located at R₁ and when it is at R₂.

Verified step by step guidance

1. Step 1: Understand the relationship between linear speed (v), angular speed (ω), and radius (r). The formula connecting these quantities is v = ω * r, where ω is the angular speed in radians per second, and r is the radius in meters.
2. Step 2: Convert the given radii R₁ and R₂ from centimeters to meters. Use the conversion factor 1 cm = 0.01 m. For example, R₁ = 2.5 cm = 0.025 m and R₂ = 5.8 cm = 0.058 m.
3. Step 3: Rearrange the formula v = ω * r to solve for angular speed ω. The formula becomes ω = v / r. Substitute the given linear speed v = 1.25 m/s and the respective radii (R₁ and R₂) into this formula to calculate the angular speeds ω₁ and ω₂ at the two radii.
4. Step 4: Convert the angular speeds ω₁ and ω₂ from radians per second to revolutions per minute (rpm). Use the conversion factors: 1 revolution = 2π radians and 1 minute = 60 seconds. The formula for conversion is ƒ = (ω * 60) / (2π). Apply this formula to both ω₁ and ω₂ to find the rotational frequencies ƒ₁ and ƒ₂ in rpm.
5. Step 5: Summarize the results. ƒ₁ corresponds to the rotational frequency when the laser is at radius R₁, and ƒ₂ corresponds to the rotational frequency when the laser is at radius R₂. These values represent how the CD player's rotational speed adjusts as the laser moves outward.

Key Concepts

Rotational Frequency

Rotational frequency, often denoted as ƒ, refers to the number of complete rotations a rotating object makes in a unit of time, typically measured in revolutions per minute (rpm). In the context of a CD player, the rotational frequency must change as the laser moves outward along the spiral track to maintain a constant linear speed of the laser beam. This adjustment ensures that the data is read accurately regardless of the position of the laser on the disc.

Linear Speed

Linear speed is the distance traveled per unit of time, expressed in meters per second (m/s). For a CD player, maintaining a constant linear speed of 1.25 m/s means that as the laser moves outward from the center of the disc, the rotational speed of the CD must decrease. This relationship between linear speed and rotational frequency is crucial for ensuring that the data encoded on the disc is read correctly.

Radius and Circumference Relationship

The relationship between radius and circumference is fundamental in circular motion, where the circumference (C) of a circle is given by the formula C = 2πr, with r being the radius. As the laser moves from radius R₁ to R₂ on the CD, the circumference of the path it follows increases, which affects the rotational frequency needed to maintain the constant linear speed. Understanding this relationship is essential for calculating the required rotational frequency at different radii.